Chapter Eight

ELECTROMAGNETIC

WAVES

8.1 INTRODUCTION

In Chapter 4, we learnt that an electric current produces magnetic field

and that two current-carrying wires exert a magnetic force on each other.

Further, in Chapter 6, we have seen that a magnetic field changing with

time gives rise to an electric field. Is the converse also true? Does an

electric field changing with time give rise to a magnetic field? James Clerk

Maxwell (1831-1879), argued that this was indeed the case – not only

an electric current but also a time-varying electric field generates magnetic

field. While applying the Ampere’s circuital law to find magnetic field at a

point outside a capacitor connected to a time-varying current, Maxwell

noticed an inconsistency in the Ampere’s circuital law. He suggested the

existence of an additional current, called by him, the displacement

current to remove this inconsistency.

Maxwell formulated a set of equations involving electric and magnetic

fields, and their sources, the charge and current densities. These

equations are known as Maxwell’s equations. Together with the Lorentz

force formula (Chapter 4), they mathematically express all the basic laws

of electromagnetism.

The most important prediction to emerge from Maxwell’s equations

is the existence of electromagnetic waves, which are (coupled) time-

varying electric and magnetic fields that propagate in space. The speed

of the waves, according to these equations, turned out to be very close to

2020-21

Physics

270

the speed of light( 3 ×10

m/s), obtained from optical

measurements. This led to the remarkable conclusion

that light is an electromagnetic wave. Maxwell’s work

thus unified the domain of electricity, magnetism and

light. Hertz, in 1885, experimentally demonstrated the

existence of electromagnetic waves. Its technological use

by Marconi and others led in due course to the

revolution in communication that we are witnessing

today.

In this chapter, we first discuss the need for

displacement current and its consequences. Then we

present a descriptive account of electromagnetic waves.

The broad spectrum of electromagnetic waves,

stretching from γ rays (wavelength ~10

–12

m) to long

radio waves (wavelength ~10

m) is described. How the

electromagnetic waves are sent and received for

communication is discussed in Chapter 15.

8.2 DISPLACEMENT CURRENT

We have seen in Chapter 4 that an electrical current

produces a magnetic field around it. Maxwell showed

that for logical consistency, a changing electric field must

also produce a magnetic field. This effect is of great

importance because it explains the existence of radio

waves, gamma rays and visible light, as well as all other

forms of electromagnetic waves.

To see how a changing electric field gives rise to

a magnetic field, let us consider the process of

charging of a capacitor and apply Ampere’s circuital

law given by (Chapter 4)

“B

dl =

i (t) (8.1)

to find magnetic field at a point outside the capacitor.

Figure 8.1(a) shows a parallel plate capacitor C which

is a part of circuit through which a time-dependent

current i (t) flows . Let us find the magnetic field at a

point such as P, in a region outside the parallel plate

capacitor. For this, we consider a plane circular loop of

radius r whose plane is perpendicular to the direction

of the current-carrying wire, and which is centred

symmetrically with respect to the wire [Fig. 8.1(a)]. From

symmetry, the magnetic field is directed along the

circumference of the circular loop and is the same in

magnitude at all points on the loop so that if B is the

magnitude of the field, the left side of Eq. (8.1) is B (2

r).

So we have

B (2πr) =

i (t) (8 .2)

JAMES CLERK MAXWELL (1831–1879)

James Clerk Maxwell

(1831 – 1879) Born in

Edinburgh, Scotland,

was among the greatest

physicists of the

nineteenth century. He

derived the thermal

velocity distribution of

molecules in a gas and

was among the first to

obtain reliable

estimates of molecular

parameters from

measurable quantities

like viscosity, etc.

Maxwell’s greatest

acheivement was the

unification of the laws of

electricity and

magnetism (discovered

by Coulomb, Oersted,

Ampere and Faraday)

into a consistent set of

equations now called

Maxwell’s equations.

From these he arrived at

the most important

conclusion that light is

an electromagnetic

wave. Interestingly,

Maxwell did not agree

with the idea (strongly

suggested by the

Faraday’s laws of

electrolysis) that

electricity was

particulate in nature.

2020-21

271

Electromagnetic

Waves

Now, consider a different surface, which has the same boundary. This

is a pot like surface [Fig. 8.1(b)] which nowhere touches the current, but

has its bottom between the capacitor plates; its mouth is the circular

loop mentioned above. Another such surface is shaped like a tiffin box

(without the lid) [Fig. 8.1(c)]. On applying Ampere’s circuital law to such

surfaces with the same perimeter, we find that the left hand side of

Eq. (8.1) has not changed but the right hand side is zero and not

since no current passes through the surface of Fig. 8.1(b) and (c). So we

have a contradiction; calculated one way, there is a magnetic field at a

point P; calculated another way, the magnetic field at P is zero.

Since the contradiction arises from our use of Ampere’s circuital law,

this law must be missing something. The missing term must be such

that one gets the same magnetic field at point P, no matter what surface

is used.

We can actually guess the missing term by looking carefully at

Fig. 8.1(c). Is there anything passing through the surface S between the

plates of the capacitor? Yes, of course, the electric field! If the plates of the

capacitor have an area A, and a total charge Q, the magnitude of the

electric field E between the plates is (Q/A)/ε

(see Eq. 2.41). The field is

perpendicular to the surface S of Fig. 8.1(c). It has the same magnitude

over the area A of the capacitor plates, and vanishes outside it. So what

is the electric flux

through the surface S ? Using Gauss’s law, it is

0 0

= =

Q Q

A A

ε ε

(8.3)

Now if the charge Q on the capacitor plates changes with time, there is a

current i = (dQ/dt), so that using Eq. (8.3), we have

t t

Q Q













ε ε

0 0

This implies that for consistency,













= i (8.4)

This is the missing term in Ampere’s circuital law. If we generalise

this law by adding to the total current carried by conductors through

the surface, another term which is ε

times the rate of change of electric

flux through the same surface, the total has the same value of current i

for all surfaces. If this is done, there is no contradiction in the value of B

obtained anywhere using the generalised Ampere’s law. B at the point P

is non-zero no matter which surface is used for calculating it. B at a

point P outside the plates [Fig. 8.1(a)] is the same as at a point M just

inside, as it should be. The current carried by conductors due to flow of

charges is called conduction current. The current, given by Eq. (8.4), is a

new term, and is due to changing electric field (or electric displacement,

an old term still used sometimes). It is, therefore, called displacement

current or Maxwell’s displacement current. Figure 8.2 shows the electric

and magnetic fields inside the parallel plate capacitor discussed above.

The generalisation made by Maxwell then is the following. The source

of a magnetic field is not just the conduction electric current due to flowing

FIGURE 8.1 A

parallel plate

capacitor C, as part of

a circuit through

which a time

dependent current

i (t) flows, (a) a loop of

radius r, to determine

magnetic field at a

point P on the loop;

(b) a pot-shaped

surface passing

through the interior

between the capacitor

plates with the loop

shown in (a) as its

rim; (c) a tiffin-

shaped surface with

the circular loop as

its rim and a flat

circular bottom S

between the capacitor

plates. The arrows

show uniform electric

field between the

capacitor plates.

2020-21

Physics

272

charges, but also the time rate of change of electric field. More

precisely, the total current i is the sum of the conduction current

denoted by i

, and the displacement current denoted by i

(= ε

dt)). So we have

c d c

i i i i

= + = +

(8.5)

In explicit terms, this means that outside the capacitor plates,

we have only conduction current i

= i, and no displacement

current, i.e., i

= 0. On the other hand, inside the capacitor, there is

no conduction current, i.e., i

= 0, and there is only displacement

current, so that i

= i.

The generalised (and correct) Ampere’s circuital law has the same

form as Eq. (8.1), with one difference: “the total current passing

through any surface of which the closed loop is the perimeter” is

the sum of the conduction current and the displacement current.

The generalised law is

B li

d =

µ µ ε

0 0

∫

(8.6)

and is known as Ampere-Maxwell law.

In all respects, the displacement current has the same physical

effects as the conduction current. In some cases, for example, steady

electric fields in a conducting wire, the displacement current may

be zero since the electric field E does not change with time. In other

cases, for example, the charging capacitor above, both conduction

and displacement currents may be present in different regions of

space. In most of the cases, they both may be present in the same

region of space, as there exist no perfectly conducting or perfectly

insulating medium. Most interestingly, there may be large regions

of space where there is no conduction current, but there is only a

displacement current due to time-varying electric fields. In such a

region, we expect a magnetic field, though there is no (conduction)

current source nearby! The prediction of such a displacement current

can be verified experimentally. For example, a magnetic field (say at point

M) between the plates of the capacitor in Fig. 8.2(a) can be measured and

is seen to be the same as that just outside (at P).

The displacement current has (literally) far reaching consequences.

One thing we immediately notice is that the laws of electricity and

magnetism are now more symmetrical*. Faraday’s law of induction states

that there is an induced emf equal to the rate of change of magnetic flux.

Now, since the emf between two points 1 and 2 is the work done per unit

charge in taking it from 1 to 2, the existence of an emf implies the existence

of an electric field. So, we can rephrase Faraday’s law of electromagnetic

induction by saying that a magnetic field, changing with time, gives rise

to an electric field. Then, the fact that an electric field changing with

time gives rise to a magnetic field, is the symmetrical counterpart, and is

FIGURE 8.2 (a) The

electric and magnetic

fields E and B between

the capacitor plates, at

the point M. (b) A cross

sectional view of Fig. (a).

* They are still not perfectly symmetrical; there are no known sources of magnetic

field (magnetic monopoles) analogous to electric charges which are sources of

electric field.

2020-21

273

Electromagnetic

Waves

EXAMPLE 8.1

a consequence of the displacement current being a source of a magnetic

field. Thus, time- dependent electric and magnetic fields give rise to each

other! Faraday’s law of electromagnetic induction and Ampere-Maxwell

law give a quantitative expression of this statement, with the current

being the total current, as in Eq. (8.5). One very important consequence

of this symmetry is the existence of electromagnetic waves, which we

discuss qualitatively in the next section.

MAXWELL’S EQUATIONS IN VACUUM

E Ad = Q /

∫

(Gauss’s Law for electricity)

0B Ad =

∫

(Gauss’s Law for magnetism)

–d

E ld =

∫

(Faraday’s Law)

µ µ ε

0 0

B ld

∫

(Ampere – Maxwell Law)

Example 8.1 A parallel plate capacitor with circular plates of radius

1 m has a capacitance of 1 nF. At t = 0, it is connected for charging in

series with a resistor R = 1 MΩ across a 2V battery (Fig. 8.3). Calculate

the magnetic field at a point P, halfway between the centre and the

periphery of the plates, after t = 10

–3

s. (The charge on the capacitor

at time t is q (t) = CV [1 – exp (–t/

)], where the time constant

equal to CR.)

FIGURE 8.3

Solution The time constant of the CR circuit is τ = CR = 10

–3

s. Then,

we have

q(t) = CV [1 – exp (–t/τ)]

= 2 × 10

–9

[1– exp (–t/10

–3

)]

The electric field in between the plates at time t is

(

)

0 0

q t

ε ε

= =

; A = π

(1)

= area of the plates.

Consider now a circular loop of radius (1/2) m parallel to the plates

passing through P. The magnetic field B at all points on the loop is

2020-21

Physics

274

along the loop and of the same value.

The flux

through this loop is

= E × area of the loop

E q

× ×













= =π

π1

2 4 4

The displacement current

(

)

–6

1 d

0.5 10 exp –1

4 d

= = ×

at t = 10

–3

s. Now, applying Ampere-Maxwell law to the loop, we get

B i i i

c d d

× ×













= +

( )

= +

( )

0 0

µ µ

= 0.5×10

–6

exp(–1)

or, B = 0.74 × 10

–13

8.3 ELECTROMAGNETIC WAVES

8.3.1 Sources of electromagnetic waves

How are electromagnetic waves produced? Neither stationary charges

nor charges in uniform motion (steady currents) can be sources of

electromagnetic waves. The former produces only electrostatic fields, while

the latter produces magnetic fields that, however, do not vary with time.

It is an important result of Maxwell’s theory that accelerated charges

radiate electromagnetic waves. The proof of this basic result is beyond

the scope of this book, but we can accept it on the basis of rough,

qualitative reasoning. Consider a charge oscillating with some frequency.

(An oscillating charge is an example of accelerating charge.) This

produces an oscillating electric field in space, which produces an oscillating

magnetic field, which in turn, is a source of oscillating electric field, and

so on. The oscillating electric and magnetic fields thus regenerate each

other, so to speak, as the wave propagates through the space.

The frequency of the electromagnetic wave naturally equals the

frequency of oscillation of the charge. The energy associated with the

propagating wave comes at the expense of the energy of the source – the

accelerated charge.

From the preceding discussion, it might appear easy to test the

prediction that light is an electromagnetic wave. We might think that all

we needed to do was to set up an ac circuit in which the current oscillate

at the frequency of visible light, say, yellow light. But, alas, that is not

possible. The frequency of yellow light is about 6 × 10

Hz, while the

frequency that we get even with modern electronic circuits is hardly about

Hz. This is why the experimental demonstration of electromagnetic

wave had to come in the low frequency region (the radio wave region), as

in the Hertz’s experiment (1887).

Hertz’s successful experimental test of Maxwell’s theory created a

sensation and sparked off other important works in this field. Two

important achievements in this connection deserve mention. Seven years

after Hertz, Jagdish Chandra Bose, working at Calcutta (now Kolkata),

EXAMPLE 8.1

2020-21

275

Electromagnetic

Waves

succeeded in producing and observing electromagnetic

waves of much shorter wavelength (25 mm to 5 mm).

His experiment, like that of Hertz’s, was confined to the

laboratory.

At around the same time, Guglielmo Marconi in Italy

followed Hertz’s work and succeeded in transmitting

electromagnetic waves over distances of many kilometres.

Marconi’s experiment marks the beginning of the field of

communication using electromagnetic waves.

8.3.2 Nature of electromagnetic waves

It can be shown from Maxwell’s equations that electric

and magnetic fields in an electromagnetic wave are

perpendicular to each other, and to the direction of

propagation. It appears reasonable, say from our

discussion of the displacement current. Consider

Fig. 8.2. The electric field inside the plates of the capacitor

is directed perpendicular to the plates. The magnetic

field this gives rise to via the displacement current is

along the perimeter of a circle parallel to the capacitor

plates. So B and E are perpendicular in this case. This

is a general feature.

In Fig. 8.4, we show a typical example of a plane

electromagnetic wave propagating along the z direction

(the fields are shown as a function of the z coordinate,

at a given time t). The electric field E

is along the x-axis,

and varies sinusoidally with z, at a given time. The

magnetic field B

is along the y-axis, and again varies

sinusoidally with z. The electric and magnetic fields E

and B

are perpendicular to each other, and to the

direction z of propagation. We can write E

and B

follows:

= E

sin (kz–

t) [8.7(a)]

= B

sin (kz–

t) [8.7(b)]

Here k is related to the wave length

of the wave by the

usual equation

(8.8)

and

is the angular frequency.

k is the magnitude of the wave

vector (or propagation vector) k

and its direction describes the

direction of propagation of the

wave. The speed of propagation

of the wave is (

/k). Using

Eqs. [8.7(a) and (b)] for E

and B

and Maxwell’s equations, one

finds that

Heinrich Rudolf Hertz

(1857 – 1894) German

physicist who was the

first to broadcast and

receive radio waves. He

produced electro-

magnetic waves, sent

them through space, and

measured their wave-

length and speed. He

showed that the nature

of their vibration,

reflection and refraction

was the same as that of

light and heat waves,

establishing their

identity for the first time.

He also pioneered

research on discharge of

electricity through gases,

and discovered the

photoelectric effect.

HEINRICH RUDOLF HERTZ (1857–1894)

FIGURE 8.4 A linearly polarised electromagnetic wave,

propagating in the z-direction with the oscillating electric field E

along the x-direction and the oscillating magnetic field B along

the y-direction.

2020-21

Physics

276

= ck, where, c = 1/

0 0

µ ε

[8.9(a)]

The relation ω = ck is the standard one for waves (see for example,

Section 15.4 of class XI Physics textbook). This relation is often written

in terms of frequency,

/2π) and wavelength,

(=2π/k) as













νλ

= c [8.9(b)]

It is also seen from Maxwell’s equations that the magnitude of the

electric and the magnetic fields in an electromagnetic wave are related as

= (E

/c) (8.10)

We here make remarks on some features of electromagnetic waves.

They are self-sustaining oscillations of electric and magnetic fields in free

space, or vacuum. They differ from all the other waves we have studied

so far, in respect that no material medium is involved in the vibrations of

the electric and magnetic fields. Sound waves in air are longitudinal waves

of compression and rarefaction. Transverse elastic (sound) waves can

also propagate in a solid, which is rigid and that resists shear. Scientists

in the nineteenth century were so much used to this mechanical picture

that they thought that there must be some medium pervading all space

and all matter, which responds to electric and magnetic fields just as any

elastic medium does. They called this medium ether. They were so

convinced of the reality of this medium, that there is even a novel called

The Poison Belt by Sir Arthur Conan Doyle (the creator of the famous

detective Sherlock Holmes) where the solar system is supposed to pass

through a poisonous region of ether! We now accept that no such physical

medium is needed. The famous experiment of Michelson and Morley in

1887 demolished conclusively the hypothesis of ether. Electric and

magnetic fields, oscillating in space and time, can sustain each other in

vacuum.

But what if a material medium is actually there? We know that light,

an electromagnetic wave, does propagate through glass, for example. We

have seen earlier that the total electric and magnetic fields inside a

medium are described in terms of a permittivity

and a magnetic

permeability

(these describe the factors by which the total fields differ

from the external fields). These replace

and

in the description to

electric and magnetic fields in Maxwell’s equations with the result that in

a material medium of permittivity

and magnetic permeability

, the

velocity of light becomes,

µε

(8.11)

Thus, the velocity of light depends on electric and magnetic properties of

the medium. We shall see in the next chapter that the refractive index of

one medium with respect to the other is equal to the ratio of velocities of

light in the two media.

The velocity of electromagnetic waves in free space or vacuum is an

important fundamental constant. It has been shown by experiments on

electromagnetic waves of different wavelengths that this velocity is the

Simulate propagation of electromagnetic waves

(i) http://www.amanogawa.com/waves.html

(ii) http://www.phys.hawaii.edu/~teb/java/ntnujava/emWave/emWave.html

2020-21

277

Electromagnetic

Waves

same (independent of wavelength) to within a few metres per second, out

of a value of 3×10

m/s. The constancy of the velocity of em waves in

vacuum is so strongly supported by experiments and the actual value is

so well known now that this is used to define a standard of length.

Namely, the metre is now defined as the distance travelled by light in

vacuum in a time (1/c) seconds = (2.99792458 × 10

)

–1

seconds. This

has come about for the following reason. The basic unit of time can be

defined very accurately in terms of some atomic frequency, i.e., frequency

of light emitted by an atom in a particular process. The basic unit of length

is harder to define as accurately in a direct way. Earlier measurement of c

using earlier units of length (metre rods, etc.) converged to a value of about

2.9979246 × 10

m/s. Since c is such a strongly fixed number, unit of

length can be defined in terms of c and the unit of time!

Hertz not only showed the existence of electromagnetic waves, but

also demonstrated that the waves, which had wavelength ten million times

that of the light waves, could be diffracted, refracted and polarised. Thus,

he conclusively established the wave nature of the radiation. Further, he

produced stationary electromagnetic waves and determined their

wavelength by measuring the distance between two successive nodes.

Since the frequency of the wave was known (being equal to the frequency

of the oscillator), he obtained the speed of the wave using the formula

v =

νλ

and found that the waves travelled with the same speed as the

speed of light.

The fact that electromagnetic waves are polarised can be easily seen

in the response of a portable AM radio to a broadcasting station. If an

AM radio has a telescopic antenna, it responds to the electric part of the

signal. When the antenna is turned horizontal, the signal will be greatly

diminished. Some portable radios have horizontal antenna (usually inside

the case of radio), which are sensitive to the magnetic component of the

electromagnetic wave. Such a radio must remain horizontal in order to

receive the signal. In such cases, response also depends on the orientation

of the radio with respect to the station.

Do electromagnetic waves carry energy and momentum like other

waves? Yes, they do. We have seen in chapter 2 that in a region of free

space with electric field E, there is an energy density (

/2). Similarly,

as seen in Chapter 6, associated with a magnetic field B is a magnetic

energy density (B

/2µ

). As electromagnetic wave contains both electric

and magnetic fields, there is a non-zero energy density associated with

it. Now consider a plane perpendicular to the direction of propagation of

the electromagnetic wave (Fig. 8.4). If there are, on this plane, electric

charges, they will be set and sustained in motion by the electric and

magnetic fields of the electromagnetic wave. The charges thus acquire

energy and momentum from the waves. This just illustrates the fact that

an electromagnetic wave (like other waves) carries energy and momentum.

Since it carries momentum, an electromagnetic wave also exerts pressure,

called radiation pressure.

If the total energy transferred to a surface in time t is U, it can be shown

that the magnitude of the total momentum delivered to this surface (for

complete absorption) is,

(8.12)

2020-21

Physics

278

EXAMPLE

EXAMPLE 8.2

When the sun shines on your hand, you feel the energy being

absorbed from the electromagnetic waves (your hands get warm).

Electromagnetic waves also transfer momentum to your hand but

because c is very large, the amount of momentum transferred is extremely

small and you do not feel the pressure. In 1903, the American scientists

Nicols and Hull succeeded in measuring radiation pressure of

visible light and verified Eq. (8.12). It was found to be of the order of

7 × 10

–6

N/m

. Thus, on a surface of area 10 cm

, the force due to radiation

is only about 7 × 10

–9

The great technological importance of electromagnetic waves stems

from their capability to carry energy from one place to another. The

radio and TV signals from broadcasting stations carry energy. Light

carries energy from the sun to the earth, thus making life possible on

the earth.

Example 8.2 A plane electromagnetic wave of frequency

25 MHz travels in free space along the x-direction. At a particular

point in space and time, E = 6.3

V/m. What is B at this point?

Solution Using Eq. (8.10), the magnitude of B is

–8

6.3 V/m

2.1 10 T

3 10 m/s

= = ×

To find the direction, we note that E is along y-direction and the

wave propagates along x-axis. Therefore, B should be in a direction

perpendicular to both x- and y-axes. Using vector algebra, E × B should

be along x-direction. Since, (+

) × (+

) =

, B is along the z-direction.

Thus, B = 2.1 × 10

–8

Example 8.3 The magnetic field in a plane electromagnetic wave is

given by B

= (2 × 10

–7

)

T sin (0.5×10

x+1.5×10

t).

(a) What is the wavelength and frequency of the wave?

(b) Write an expression for the electric field.

Solution

(a) Comparing the given equation with

= B

sin

x t

























We get,

0.5 10

m = 1.26 cm,

and

( )

1.5 10 /2 23.9 GHz

= = × π =

(b) E

= B

c = 2×10

–7

T × 3 × 10

m/s = 6 × 10

V/m

The electric field component is perpendicular to the direction of

propagation and the direction of magnetic field. Therefore, the

electric field component along the z-axis is obtained as

= 60 sin (0.5 × 10

x + 1.5 × 10

t) V/m

2020-21

279

Electromagnetic

Waves

EXAMPLE 8.4

Example 8.4 Light with an energy flux of 18 W/cm

falls on a non-

reflecting surface at normal incidence. If the surface has an area of

20 cm

, find the average force exerted on the surface during a 30

minute time span.

Solution

The total energy falling on the surface is

U = (18 W/cm

) × (20 cm

) × (30 × 60 s)

= 6.48 × 10

Therefore, the total momentum delivered (for complete absorption) is

p =

6.48 10 J

3 10 m/s

= 2.16 × 10

–3

kg m/s

The average force exerted on the surface is

F =

2.16 10

1.2 10 N

0.18 10

−

= = ×

How will your result be modified if the surface is a perfect reflector?

Example 8.5 Calculate the electric and magnetic fields produced by

the radiation coming from a 100 W bulb at a distance of 3 m. Assume

that the efficiency of the bulb is 2.5% and it is a point source.

Solution The bulb, as a point source, radiates light in all directions

uniformly. At a distance of 3 m, the surface area of the surrounding

sphere is

2 2 2

4 4 (3) 113m

A r= π = π =

The intensity I at this distance is

100 W 2.5 %

Power

Area

113 m

= =

= 0.022 W/m

Half of this intensity is provided by the electric field and half by the

magnetic field.

( )

1 1

2 2

0.022 W/m

rms

I E c

( )( )

12 8

0.022

V/m

8.85 10 3 10

rms

−

× ×

= 2.9 V/m

The value of E found above is the root mean square value of the

electric field. Since the electric field in a light beam is sinusoidal, the

peak electric field, E

rms

2 2 2.9 V/m

E = ×

= 4.07 V/m

Thus, you see that the electric field strength of the light that you use

for reading is fairly large. Compare it with electric field strength of

TV or FM waves, which is of the order of a few microvolts per metre.

EXAMPLE 8.5

2020-21

Physics

280

EXAMPLE 8.5

Now, let us calculate the strength of the magnetic field. It is

8 1

2.9 V m

3 10 m s

rms

−

= =

= 9.6 × 10

–9

Again, since the field in the light beam is sinusoidal, the peak

magnetic field is B

rms

= 1.4 × 10

–8

T. Note that although the

energy in the magnetic field is equal to the energy in the electric

field, the magnetic field strength is evidently very weak.

8.4 ELECTROMAGNETIC

SPECTRUM

At the time Maxwell predicted the existence of electromagnetic waves, the

only familiar electro

magnetic

waves were the visible light waves. The existence

of ultraviolet and infrared waves was barely established. By the end of the

nineteenth century, X-rays and gamma rays had also been discovered. We

now know that, electro

magnetic waves include visible light waves, X-rays,

gamma rays, radio waves, microwaves, ultraviolet and infrared waves. The

classification of em waves according to frequency is the electromagnetic

spectrum (Fig. 8.5). There is no sharp division between one kind of wave

and the next. The classification is based roughly on how the waves are

produced and/or detected.

FIGURE 8.5 The electromagnetic spectrum, with common names for various

part of it. The various regions do not have sharply defined boundaries.

Electromagnetic spectrum

http://www.fnal.gov/pub/inquiring/more/light

http://imagine.gsfc.nasa.gov/docs/science/

2020-21

281

Electromagnetic

Waves

We briefly describe these different types of electromagnetic waves, in

order of decreasing wavelengths.

8.4.1 Radio waves

Radio waves are produced by the accelerated motion of charges in conducting

wires. They are used in radio and television communication systems. They

are generally in the frequency range from 500 kHz to about 1000 MHz.

The AM (amplitude modulated) band is from 530 kHz to 1710 kHz. Higher

frequencies upto 54 MHz are used for short wave bands. TV waves range

from 54 MHz to 890 MHz. The FM (frequency modulated) radio band

extends from 88 MHz to 108 MHz. Cellular phones use radio waves to

transmit voice communication in the ultrahigh frequency (UHF) band. How

these waves are transmitted and received is described in Chapter 15.

8.4.2 Microwaves

Microwaves (short-wavelength radio waves), with frequencies in the

gigahertz (GHz) range, are produced by special vacuum tubes (called

klystrons, magnetrons and Gunn diodes). Due to their short wavelengths,

they are suitable for the radar systems used in aircraft navigation. Radar

also provides the basis for the speed guns used to time fast balls, tennis-

serves, and automobiles. Microwave ovens are an interesting domestic

application of these waves. In such ovens, the frequency of the microwaves

is selected to match the resonant frequency of water molecules so that

energy from the waves is transferred efficiently to the kinetic energy of

the molecules. This raises the temperature of any food containing water.

MICROWAVE OVEN

The spectrum of electromagnetic radiation contains a part known as microwaves. These

waves have frequency and energy smaller than visible light and wavelength larger than it.

What is the principle of a microwave oven and how does it work?

Our objective is to cook food or warm it up. All food items such as fruit, vegetables,

meat, cereals, etc., contain water as a constituent. Now, what does it mean when we say that

a certain object has become warmer? When the temperature of a body rises, the energy of

the random motion of atoms and molecules increases and the molecules travel or vibrate or

rotate with higher energies. The frequency of rotation of water molecules is about

2.45 gigahertz (GHz). If water receives microwaves of this frequency, its molecules absorb

this radiation, which is equivalent to heating up water. These molecules share this energy

with neighbouring food molecules, heating up the food.

One should use porcelain vessels and not metal containers in a microwave oven because

of the danger of getting a shock from accumulated electric charges. Metals may also melt

from heating. The porcelain container remains unaffected and cool, because its large

molecules vibrate and rotate with much smaller frequencies, and thus cannot absorb

microwaves. Hence, they do not get heated up.

Thus, the basic principle of a microwave oven is to generate microwave radiation of

appropriate frequency in the working space of the oven where we keep food. This way

energy is not wasted in heating up the vessel. In the conventional heating method, the vessel

on the burner gets heated first, and then the food inside gets heated because of transfer of

energy from the vessel. In the microwave oven, on the other hand, energy is directly delivered

to water molecules which is shared by the entire food.

2020-21

Physics

282

8.4.3 Infrared waves

Infrared waves are produced by hot bodies and molecules. This band

lies adjacent to the low-frequency or long-wave length end of the visible

spectrum. Infrared waves are sometimes referred to as heat waves. This

is because water molecules present in most materials readily absorb

infrared waves (many other molecules, for example, CO

, NH

, also absorb

infrared waves). After absorption, their thermal motion increases, that is,

they heat up and heat their surroundings. Infrared lamps are used in

physical therapy. Infrared radiation also plays an important role in

maintaining the earth’s warmth or average temperature through the

greenhouse effect. Incoming visible light (which passes relatively easily

through the atmosphere) is absorbed by the earth’s surface and re-

radiated as infrared (longer wavelength) radiations. This radiation is

trapped by greenhouse gases such as carbon dioxide and water vapour.

Infrared detectors are used in Earth satellites, both for military purposes

and to observe growth of crops. Electronic devices (for example

semiconductor light emitting diodes) also emit infrared and are widely

used in the remote switches of household electronic systems such as TV

sets, video recorders and hi-fi systems.

8.4.4 Visible rays

It is the most familiar form of electromagnetic waves. It is the part of the

spectrum that is detected by the human eye. It runs from about

4 × 10

Hz to about 7 × 10

Hz or a wavelength range of about 700 –

400 nm. Visible light emitted or reflected from objects around us provides

us information about the world. Our eyes are sensitive to this range of

wavelengths. Different animals are sensitive to different range of

wavelengths. For example, snakes can detect infrared waves, and the

‘visible’ range of many insects extends well into the utraviolet.

8.4.5 Ultraviolet rays

It covers wavelengths ranging from about 4 × 10

–7

m (400 nm) down to

6 × 10

–10

m (0.6 nm). Ultraviolet (UV) radiation is produced by special

lamps and very hot bodies. The sun is an important source of ultraviolet

light. But fortunately, most of it is absorbed in the ozone layer in the

atmosphere at an altitude of about 40 – 50 km. UV light in large quantities

has harmful effects on humans. Exposure to UV radiation induces the

production of more melanin, causing tanning of the skin. UV radiation is

absorbed by ordinary glass. Hence, one cannot get tanned or sunburn

through glass windows.

Welders wear special glass goggles or face masks with glass windows

to protect their eyes from large amount of UV produced by welding arcs.

Due to its shorter wavelengths, UV radiations can be focussed into very

narrow beams for high precision applications such as LASIK (Laser-

assisted in situ keratomileusis) eye surgery. UV lamps are used to kill

germs in water purifiers.

Ozone layer in the atmosphere plays a protective role, and hence its

depletion by chlorofluorocarbons (CFCs) gas (such as freon) is a matter

of international concern.

2020-21

283

Electromagnetic

Waves

8.4.6 X-rays

Beyond the UV region of the electromagnetic spectrum lies the X-ray

region. We are familiar with X-rays because of its medical applications. It

covers wavelengths from about 10

–8

m (10 nm) down to 10

–13

(10

–4

nm). One common way to generate X-rays is to bombard a metal

target by high energy electrons. X-rays are used as a diagnostic tool in

medicine and as a treatment for certain forms of cancer. Because X-rays

damage or destroy living tissues and organisms, care must be taken to

avoid unnecessary or over exposure.

8.4.7 Gamma rays

They lie in the upper frequency range of the electromagnetic spectrum

and have wavelengths of from about 10

–10

m to less than 10

–14

m. This

high frequency radiation is produced in nuclear reactions and

also emitted by radioactive nuclei. They are used in medicine to destroy

cancer cells.

Table 8.1 summarises different types of electromagnetic waves, their

production and detections. As mentioned earlier, the demarcation

between different regions is not sharp and there are overlaps.

TABLE 8.1 DIFFERENT TYPES OF ELECTROMAGNETIC WAVES

Type Wavelength range Production Detection

Radio > 0.1 m Rapid acceleration and Receiver’s aerials

decelerations of electrons

in aerials

Microwave 0.1m to 1 mm Klystron valve or Point contact diodes

magnetron valve

Infra-red 1mm to 700 nm Vibration of atoms Thermopiles

and molecules Bolometer, Infrared

photographic film

Light 700 nm to 400 nm Electrons in atoms emit The eye

light when they move from Photocells

one energy level to a Photographic film

lower energy level

Ultraviolet 400 nm to 1nm Inner shell electrons in Photocells

atoms moving from one Photographic film

energy level to a lower level

X-rays 1nm to 10

–3

nm X-ray tubes or inner shell Photographic film

electrons Geiger tubes

Ionisation chamber

Gamma rays <10

–3

nm Radioactive decay of the -do-

nucleus

2020-21

Physics

284

SUMMARY

1. Maxwell found an inconsistency in the Ampere’s law and suggested the

existence of an additional current, called displacement current, to remove

this inconsistency. This displacement current is due to time-varying electric

field and is given by

and acts as a source of magnetic field in exactly the same way as conduction

current.

2. An accelerating charge produces electromagnetic waves. An electric charge

oscillating harmonically with frequency

, produces electromagnetic waves

of the same frequency

. An electric dipole is a basic source of

electromagnetic waves.

3. Electromagnetic waves with wavelength of the order of a few metres were

first produced and detected in the laboratory by Hertz in 1887. He thus

verified a basic prediction of Maxwell’s equations.

4. Electric and magnetic fields oscillate sinusoidally in space and time in an

electromagnetic wave. The oscillating electric and magnetic fields, E and

B are perpendicular to each other, and to the direction of propagation of

the electromagnetic wave. For a wave of frequency

, wavelength

propagating along z-direction, we have

= E

(t) = E

sin (kz –

t )

= E

sin

2 2

π π

t E

z t

−

























= −

























sin

B = B

(t) = B

sin (kz –

t B

z t

0 0

2 2sin sinπ π

−

























= −

























They are related by E

= c.

5. The speed c of electromagnetic wave in vacuum is related to

and

(the

free space permeability and permittivity constants) as follows:

0 0

1/c

µ ε

. The value of c equals the speed of light obtained from

optical measurements.

Light is an electromagnetic wave; c is, therefore, also the speed of light.

Electromagnetic waves other than light also have the same velocity c in

free space.

The speed of light, or of electromagnetic waves in a material medium is

given by

1/v

µ ε

where

is the permeability of the medium and

its permittivity.

6. Electromagnetic waves carry energy as they travel through space and this

energy is shared equally by the electric and magnetic fields.

Electromagnetic waves transport momentum as well. When these waves

strike a surface, a pressure is exerted on the surface. If total energy

transferred to a surface in time t is U, total momentum delivered to this

surface is p = U/c.

7. The spectrum of electromagnetic waves stretches, in principle, over an

infinite range of wavelengths. Different regions are known by different

2020-21

285

Electromagnetic

Waves

names; γ-rays, X-rays, ultraviolet rays, visible rays, infrared rays,

microwaves and radio waves in order of increasing wavelength from 10

–2

or 10

–12

m to 10

They interact with matter via their electric and magnetic fields which set

in oscillation charges present in all matter. The detailed interaction and

so the mechanism of absorption, scattering, etc., depend on the wavelength

of the electromagnetic wave, and the nature of the atoms and molecules

in the medium.

POINTS TO PONDER

1. The basic difference between various types of electromagnetic waves

lies in their wavelengths or frequencies since all of them travel through

vacuum with the same speed. Consequently, the waves differ

considerably in their mode of interaction with matter.

2. Accelerated charged particles radiate electromagnetic waves. The

wavelength of the electromagnetic wave is often correlated with the

characteristic size of the system that radiates. Thus, gamma radiation,

having wavelength of 10

–14

m to 10

–15

m, typically originate from an

atomic nucleus. X-rays are emitted from heavy atoms. Radio waves

are produced by accelerating electrons in a circuit. A transmitting

antenna can most efficiently radiate waves having a wavelength of

about the same size as the antenna. Visible radiation emitted by atoms

is, however, much longer in wavelength than atomic size.

3. The oscillating fields of an electromagnetic wave can accelerate charges

and can produce oscillating currents. Therefore, an apparatus designed

to detect electromagnetic waves is based on this fact. Hertz original

‘receiver’ worked in exactly this way. The same basic principle is utilised

in practically all modern receiving devices. High frequency

electromagnetic waves are detected by other means based on the

physical effects they produce on interacting with matter.

4. Infrared waves, with frequencies lower than those of visible light,

vibrate not only the electrons, but entire atoms or molecules of a

substance. This vibration increases the internal energy and

consequently, the temperature of the substance. This is why infrared

waves are often called heat waves.

5. The centre of sensitivity of our eyes coincides with the centre of the

wavelength distribution of the sun. It is because humans have evolved

with visions most sensitive to the strongest wavelengths from

the sun.

EXERCISES

8.1 Figure 8.6 shows a capacitor made of two circular plates each of

radius 12 cm, and separated by 5.0 cm. The capacitor is being

charged by an external source (not shown in the figure). The

charging current is constant and equal to 0.15A.

(a) Calculate the capacitance and the rate of change of potential

difference between the plates.

2020-21

Physics

286

(b) Obtain the displacement current across the plates.

capacitor? Explain.

FIGURE 8.6

8.2 A parallel plate capacitor (Fig. 8.7) made of circular plates each of radius

R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to

a 230 V ac supply with a (angular) frequency of 300 rad s

–1

(a) What is the rms value of the conduction current?

(b) Is the conduction current equal to the displacement current?

between the plates.

FIGURE 8.7

8.3 What physical quantity is the same for X-rays of wavelength

–10

m, red light of wavelength 6800 Å and radiowaves of wavelength

500m?

8.4 A plane electromagnetic wave travels in vacuum along z-direction.

What can you say about the directions of its electric and magnetic

field vectors? If the frequency of the wave is 30 MHz, what is its

wavelength?

8.5 A radio can tune in to any station in the 7.5 MHz to 12 MHz band.

What is the corresponding wavelength band?

8.6 A charged particle oscillates about its mean equilibrium position

with a frequency of 10

Hz. What is the frequency of the

electromagnetic waves produced by the oscillator?

8.7 The amplitude of the magnetic field part of a harmonic

electromagnetic wave in vacuum is B

= 510 nT. What is the

amplitude of the electric field part of the wave?

8.8 Suppose that the electric field amplitude of an electromagnetic wave

is E

= 120 N/C and that its frequency is

= 50.0 MHz. (a) Determine,

,ω, k, and

. (b) Find expressions for E and B.

8.9 The terminology of different parts of the electromagnetic spectrum

is given in the text. Use the formula E = h

(for energy of a quantum

of radiation: photon) and obtain the photon energy in units of eV for

different parts of the electromagnetic spectrum. In what way are

the different scales of photon energies that you obtain related to the

sources of electromagnetic radiation?

8.10 In a plane electromagnetic wave, the electric field oscillates

sinusoidally at a frequency of 2.0 × 10

Hz and amplitude 48 V m

–1

2020-21

287

Electromagnetic

Waves

(a) What is the wavelength of the wave?

(b) What is the amplitude of the oscillating magnetic field?

average energy density of the B field. [c = 3 × 10

m s

–1

ADDITIONAL EXERCISES

8.11 Suppose that the electric field part of an electromagnetic wave in

vacuum is E = {(3.1 N/C) cos [(1.8 rad/m) y + (5.4 × 10

rad/s)t]}

(a) What is the direction of propagation?

(b) What is the wavelength

(d) What is the amplitude of the magnetic field part of the wave?

(e) Write an expression for the magnetic field part of the wave.

8.12 About 5% of the power of a 100 W light bulb is converted to visible

radiation. What is the average intensity of visible radiation

(a) at a distance of 1m from the bulb?

(b) at a distance of 10 m?

Assume that the radiation is emitted isotropically and neglect

reflection.

8.13 Use the formula

T = 0.29 cmK to obtain the characteristic

temperature ranges for different parts of the electromagnetic

spectrum. What do the numbers that you obtain tell you?

8.14 Given below are some famous numbers associated with

electromagnetic radiations in different contexts in physics. State

the part of the electromagnetic spectrum to which each belongs.

(a) 21 cm (wavelength emitted by atomic hydrogen in interstellar

space).

(b) 1057 MHz (frequency of radiation arising from two close energy

levels in hydrogen; known as Lamb shift).

all space-thought to be a relic of the ‘big-bang’ origin of the

universe].

(d) 5890 Å - 5896 Å [double lines of sodium]

(e) 14.4 keV [energy of a particular transition in

Fe nucleus

associated with a famous high resolution spectroscopic method

(Mössbauer spectroscopy)].

8.15 Answer the following questions:

(a) Long distance radio broadcasts use short-wave bands. Why?

(b) It is necessary to use satellites for long distance TV transmission.

Why?

astronomy is possible only from satellites orbiting the earth.

Why?

(d) The small ozone layer on top of the stratosphere is crucial for

human survival. Why?

(e) If the earth did not have an atmosphere, would its average

surface temperature be higher or lower than what it is now?

(f) Some scientists have predicted that a global nuclear war on the

earth would be followed by a severe ‘nuclear winter’ with a

devastating effect on life on earth. What might be the basis of

this prediction?

2020-21